Given three planes in space, a complete characterization of their intersection is provided. Special attention is paid to the case when the intersection set does not exist of one point only. Besides the vector cross product, the tool of generalized inverse of a matrix is used extensively.

In a previous study, the onto-semiotic approach was employed to analyse the mathematical notion of different coordinate systems, as well as some situations and university students' actions related to these coordinate systems in the context of multivariate calculus. This study approaches different coordinate systems through the process of change of…

Most students complete their first and only course in linear algebra with the understanding that a real, square matrix "A" has an inverse if and only if "rref"("A"), the reduced row echelon form of "A", is the identity matrix I[subscript n]. That is, if they apply elementary row operations via the Gauss-Jordan algorithm to the partitioned matrix…

This paper describes a case study of two pure mathematicians who flipped their lecture to teach matrix determinants in two large mathematics service courses (one at Stage I and the other at Stage II). The purpose of the study was to transform the passive lecture into an active learning opportunity and to introduce valuable mathematical skills,…

When we use technology to teach mathematics, we hope to focus on the mathematics, restricting the computer software systems to providing support for our pedagogy. It is a matter of common experience, however, that students can become distracted or frustrated by the quirks of the particular software system being used. Here, experience using the…

In this study, I considered the development of mathematical meaning related to the Invertible Matrix Theorem (IMT) for both a classroom community and an individual student over time. In this particular linear algebra course, the IMT was a core theorem in that it connected many concepts fundamental to linear algebra through the notion of…

The material in this module introduces students to some of the mathematical tools used to examine molecular evolution. This topic is standard fare in many mathematical biology or bioinformatics classes, but could also be suitable for classes in linear algebra or probability. While coursework in matrix algebra, Markov processes, Monte Carlo…

Benford's law of the "first digits" states that in spite of intuitively expected equal frequency of 1/9 of the decimal digits r = 1, ... , 9 appearance on the first place of any number, various empirical studies show another pattern of these frequencies distribution, which is log[subscript 10](1 + 1/r). The article considers this law and other…

The use of digital elevation models, which represent the surface of the earth by a matrix of heights, has proved an ideal introductory topic for developing both the matrix-handling and computational skills of undergraduate mathematicians. This article looks at the solving of an easily understandable, but not necessarily simple problem: given a…

Characteristic polynomials are used to determine when magic squares have magic inverses. A resulting method constructs arbitrary examples of such squares.

The life and work of J. J. Sylvester, a mathematician who developed the theory of determinants, are briefly described. (SD)

For two given vectors of the three-dimensional Euclidean space we investigate the problem of identifying all rotations that transform them into each other. For this purpose we consider three types of rotation matrices to obtain a complete characterization. Finally some attention is paid to the problem of obtaining all rotations taking two vectors…

An easy way to present the uniqueness of the square root of a positive semidefinite matrix is given.